1. Introduction

The concept of multiplicative metric spaces is introduced by M. Özavsar ^[8].They also gave some topological properties of the relevant multiplicative metric space and now it’s more general than well-known metric space. Fixed point theorems are admirable tool for Existence and uniqueness of the solutions to various mathematical models like differential, integral and partial differential equations and vibrational inequalities etc. The study of metric space plays very important role to many fields both in pure and applied science ^[4]. Abounding researchers extended the notion of a metric space such as vector valued metric space of Perov ^[3], a cone metric spaces of Huang and Zhang ^[7], a modular metric spaces of Chistyakov ^[17], for details about multiplicative metric space and related concepts, we refer the reader to^[8] etc.

It is well know that the set of positive real numbers is not complete according to the usual metric. To overcome this problem, In 2008, Bashirov ^[2] Introduced the concept of multiplicative metric spaces as follows:

Definition 1.1. ^[8] Let X be a nonempty set. Multiplicative metric is a mapping d: XX → satisfying the following conditions

(1.1) d(x, y) 1 for all x, y X and d(x, y) = 1 if and only if x = y,

(1.2) d(x, y) = d(y, x) for all x, y X,

(1.3) d(x, z) ≤ d(x, y) ∙d(y, z) for all x, y, z X (multiplicative triangle inequality)

To articulate the importance of this study, we should first note that is a complete multiplicative metric space with respect to the multiplicative metric. Furthermore, we introduce concept of multiplicative contraction mapping and prove some fixed point theorems of multiplicative contraction mappings on multiplicative metric spaces.

Definition 1.2. ^[8] (Multiplicative ball) Let (X, d) be a multiplicative metric space, x X and > 1. We now define a set = {y X | d(x, y) < }, which is called multiplicative open ball of radius with centre x. Similarly, one can describe multiplicative closed ball as

Definition 1.3. ^[8] (Multiplicative interior point): Let (X, d) be a multiplicative metric space and A X. Then we call x A a multiplicative interior point of A if there exists an > 1 such that A. The collection of all interior points of A is called multiplicative interior of A and denoted by int(A).

Definition 1.4. ^[8] (Multiplicative open set): Let (X, d) be a multiplicative metric space and A X. If every point of A is a multiplicative interior point of A, i.e., A = int(A), then A is called a multiplicative open set.

Definition 1.5. ^[8] Let (X, d) be a multiplicative metric space. A point x X is said to be multiplicative limit point of S X if and only if ( \ {x}) ∩ S ∅ for every > 1. The set of all multiplicative limit points of the set S is denoted by S ′.

Definition 1.6. ^[8] Let (X, d) be a multiplicative metric space. We call a set S X multiplicative closed in (X, d) if S contains all of its multiplicative limit points.

Definition 1.7. ^[8] (Multiplicative convergence): Let (X, d) be a multiplicative metric space, {x_n} be a sequence in X and x X. If for every multiplicative open ball , there exists a natural number N such that n ≥ N⇒x_n, then the sequence {x_n} is said to be multiplicative convergent to x, denoted by x_n → x (n → ∞).

Lemma 1.8. ^[8] Let (X, d) be a multiplicative metric space, {x_n} be a sequence in X and x X. Then x_n →x (n → ∞) if and only if d(x _n, x) → 1 (n → ∞).

Lemma 1.9. ^[8] Let (X, d) be a multiplicative metric space, {x_n} be a sequence in X. If the sequence {x_n} is multiplicative convergent, then the multiplicative limit point is unique.

Definition 1.10. ^[8] Let (X, d) be a multiplicative metric space and {x_n} be a sequence in X. The sequence is called a multiplicative Cauchy sequence if it holds that, for all > 1, there exists such that d(x_m, x_n) < for m, n ≥ N.

Definition 1.11. ^[8] Let (X, d) be a multiplicative metric space and A X. The set A is called multiplicative bounded if there exist and M > 1 such that A B_M(x).

Lemma 1.12. ^[8] Let (X, d) be a multiplicative metric space and {x_n} be a sequence in X. Then {x_n} is a multiplicative Cauchy sequence if and only if d(x_n, x_m) →1 (m, n → ∞).

Definition 1.13. ^[5] Let S and T be self-maps of multiplicative metric space a non-empty set X. then

i. Any point is said to be fixed point of T if Tx=x.

ii. Any point is said to be coincidence point of T and S if Sx=Tx and we shall called w=Sx=Tx that a point of coincidence of S and T.

iii. Any point is said to be fixed point of T and S if Sx=Tx=x

Definition 1.14. ^[14] Let S and T be self-maps of multiplicative metric space (X, d), then S,T are said to be weakly commuting if d(STx, TSx)d(Sx,Tx), for all

Definition 1.15. ^[5] Two self-maps of multiplicative metric space S, T of a non-empty set X are said to be commuting is TSx=STx for all

Definition 1.16. ^[5] Let S, T be self-maps of multiplicative metric space (X,d), then S,T are said to be compatible if , Whenever {x_n} is a sequence in X such that for some

Definition 1.17. ^[5] Two self-maps of multiplicative metric space S, T of a non-empty set X are said to be weakly compatible is STx = TSx whenever Sx = Tx.

2. Main Results

Theorem 2.1 let (X,d) be a complete multiplicative metric space and P,Q,R,S,T,U be self-maps of X satisfying the following condition

(2.1)

(2.2)

for all x, y X, is a constant. Assume that the pairs (TU, Q), (RS, P) are weakly compatible. Pairs (T, U), (T, Q), (U, Q), (R, S), (R, P), (S, P) are commuting pairs of maps. Then P, Q, R, S, T, U have a unique common fixed point in X.

Proof. Let by (2.1) we can define inductively a sequence {y_n} in X such that and for all n=0,1, 2, 3 …

By (2.1), we have

(2.3)

Similarly, we have

(2.4)

where h =<1 as

Therefore, using (2.3) and (2.4), we have

(2.5)

for n=0,1, 2, 3…

Now, for all m > n

which implies that, as . Hence is a Cauchy sequence, by the completeness of X, there exist such that,

(2.6)

Since, TU(X)P(X) there exist such , we claim that , if possible , if possible then by using (2.2), we have

taking limit as n, we have

which is a contradiction. Therefore,

Since, RS(X)Q(X) there exist v such that .

We claim that TUv=z, if possible TUv, then by using (2.2), we have

we have,

(2.8)

which is a contradiction.

Therefore, TUv=Qv=z.

Here, Q and TU are weakly compatibles, we have TUz=Qz.

Now we claim that z is a fixed point of TU. If TUz≠z, then by (2.2), we have

therefore,

which is a contradiction.

Therefore, TUz=z, hence Qz=z. so we have

(2.9)

So, z is a common fixed point of TU and Q.

Similarly, P and RS are weakly compatible maps, we have RSz=Pz.

Now we claim that z is a fixed point of RS. If RSz≠z, then by (2.2), we have

we have,

which is a contradiction.

Therefore, RSz = z, hencePz = z. so, we have

(2.10)

So z is a common fixed point of TU, Q, P and RS.

By commuting condition of pairs

which follows that, Tz and Uz are common fixed points of TU and P, then

(2.11)

Similarly, by commuting property,

which follows that, Rz and Sz are common fixed points of RS and Q, then

Then Rz = z = Sz = Qz = RSz.

Therefore z is a common fixed point of T, U, R, S, P and Q.

Let w be other common fixed point of T, U, R, P, S and Q. if possible w, we have

a contradiction.

So z = w.

In Theorem 2.1, if we put S = U= 1, then we obtain the following corollary.

Corollary 2.2. Let (X, d) be a complete multiplicative metric space and P,Q,R ,T be self-maps of X satisfying the following condition

(2.13)

(2.14)

for all x, y X, is a constant. Assume that the pairs (T, Q), (R, P) are weakly compatible. Pairs (T,Q), (U,Q), (R,P), (S,P) are commuting pairs of maps. Then P, Q, R, T have a unique common fixed point in X.

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